four square theorem - quaternions

8/12/2019 Four Square Theorem - Quaternions

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QUARTERNIONS AND THE FOUR SQUARE THEOREM

JIA HONG RAY NG

Abstract. The Four Square Theorem was proved by Lagrange in 1770: ev-ery positive integer is the sum of at most four squares of positive integers, i.e.

n= A2 +B2 +C2 +D2,A,B,C,D Z An interesting proof is presented herebased on Hurwitz integers, a subset of quarternions which act like integersin four dimensions and have the prime divisor property. It is analogous to

Fermats Two Square Theorem, which says that positive primes of the form4k+1 can be written as the sum of the squares of two positive integers. Rep-

resenting integers as the sum of squares can be considered a special case ofWarings problem: for every k is there a numberg (k) such that every integer

is representable by at mostg (k) kth

powers?

Contents

1. Quarternions and Hurwitz Integers 12. Four Square Theorem 43. Discussion 6References 6

1. Quarternions and Hurwitz Integers

Definition 1.1. For each pair, C, a quarternion is a four dimensional number

represented by the matrix q=

.

The set of quarternions is called H, after its discoverer Hamiltion (1843). If = a+di and = b+ci, where a,b,c,d R, then each quarternion is a linearcombination of the quarternion units 1,i,j,k.

=

a+di b+cib+ci a di

= a

1 00 1

+b

0 11 0

+c

0 ii 0

+d

i 00 i

= a1+bi+cj+dk

1


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2 JIA HONG RAY NG

Definition 1.2. The norm of a quarternion, q is defined to be the square rootof its determinant, hence q2 is

det

= += ||2 + ||2

det

a +di b+cib+ci a di

=a2 +b2 +c2 +d2

The quarternions 1, i, j, k have norm 1 and satisfy the following relations:

i2 =j2 =k2 =1,

ij= k =ji,

jk= i = kj,

ki= j =ik

Non-zero quarternions form a group under multiplication, but the product ofquarternions is generally non-commutative: q1q2 = q2q1. Quarternions also form

an abelian group under addition, and obey the left and right distributive laws:q1 (q2+q3) = q1q2+q1q3, (q2+q3) = q2q1+q3q1.

Due to the multiplicative property of the determinants,det (q1) det (q2) = det (q1q2),the norm also has a multiplicative property, i.e.

norm (q1) norm (q2) = norm (q1q2) .

It is interesting to note that n = 1, 2, 4, 8 are the only n for which Rn has amultiplication that distributes over vector addition, and a multiplicative norm.They correspond to R,C,H and O (the octonions, n = 8) respectively.

Theorem 1.3. Four Square Identity: If two numbers can each be written as thesum of four squares, then so can their product.

Proof. Let x1= a21+b

21+c

21+d

21 = a11 +b1i +c1j +d1k

2 =q12 and

x2 = a22+b22+c22+d22 = a21 +b2i +c2j +d2k2 =q22Then by the multiplicative property of norms,x1x2 = q12q22 =q1q22

Therefore,x1x2 = a2 +b2 +c2 +d2, where q1q2

2 =a2 +b2 +c2 +d2

Definition 1.4. The conjugate of any quarternion q = a+ bi+ cj+ dk is q =a bi cj dk. Conjugation has the following easily-proved properties:

qq= |q|2,

q1+q2 = q1+q2,

q1 q2 = q1 q2,

q1q2= q2 q1

(due to non-commutative quarternion multiplication).

Definition 1.5. The Hurwitz integers are quarternions that make up the setof all the integer combinations of 1+i+j+k

2 , i,j, k, denoted by Z [h, i,j, k], where

h = 1+i+j+k2

. It contains the set Z [i,j, k] = {a+bi +cj +dk: a, b, c, d Z}.There are 24 units (that form a non-abelian group): the 8 units 1, i, j + kof

Z [i,j, k], and the 16 mid-points 12

i2

j2

k2

. The Hurwitz integers are closedunder addition and multiplication and the square of the norm is always an ordinaryinteger.


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QUARTERNIONS AND THE FOUR SQUARE THEOREM 3

They also obey the division property, which says:If , = 0 are in Z [h, i,j, k], then there is a quotient and a remainder suchthat = + with ||


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Proof. Assume that the primep dividesbut does not divide . Then accordingto Theorem 1.8,

1 = right gcd(p, ) = p+.Multiplying on the right by for both sides,

= p+.

Note thatp is both a right and left divisor of whatever number it divides, since realscommute with quarternions. Thereforepdivides bothp, and(by assumption).Consequently, p divides , as required by the statement.

2. Four Square Theorem

Now, we can use the above properties of quarternions to prove the Four SquareTheorem, which says thatevery natural number is the sum of four squares. In fact,the Four Square Identity (Theorem 1.3) implies that if every prime is the sum offour squares, then all the other natural numbers except 1, which are products ofprimes, can satisfy the Four Square Theorem. Clearly, 1 = 02 + 02 + 02 + 12 and2 = 02 + 02 + 12 + 12, therefore we can immediately simplify the problem to that ofrepresenting every odd prime p as the sum of four integer squares. The followingproof is due to Hurwitz himself, and differs from Lagranges own proof, which arguesthat the least integer m satisfying the equation mp = A2 +B2 +C2 +D2 is 1.

Theorem 2.1. (Conditional Four Square Theorem): any ordinary prime p that isnot a Hurwitz prime is a sum of four integer squares.

Proof. Suppose p has a nontrivial Hurwitz integer factorization

p= (a+bi +cj +dk)

Taking conjugates on both sides,

p= p = (a bi cj dk)

Multiplying both expressions,

p2 = (a+bi +cj +dk) (a bi cj dk)

= (a+bi +cj +dk) (a bi cj dk)

=

a2 +b2 +c2 +d2

||2

where both a2 +b2 +c2 +d2 and ||2 are integers greater than 1. However, byunique prime factorization, the only positve integer factorization is pp, therefore

p= a2 + b2 + c2 + d2 = (a+bi +cj +dk) (a bi cj dk). If the Hurwitz integera + bi + cj + dkis also in the set Z [i,j, k], then p is the sum of four integer squares.

But suppose = a + bi + cj + dkhas half-integer co-ordinates, the factors ofp2

can be re-expressed to achieve integer values for the co-efficients of 1, i,j, k in .Consider the integer = 1ijk

2 , which has norm 1, thus = 1. By choosing

appropriate signs in , can be written as = + a +bi+ cj+ dk, wherea, b, c, d are even integers. Therefore,

p= (a+bi +cj +dk) (a bi cj dk)

= (+a +bi +cj +dk) (+a bi cj dk)

= (+a +bi +cj +dk) (+a bi cj dk)


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QUARTERNIONS AND THE FOUR SQUARE THEOREM 5

With respect to the first factor, the product of and plus the even terms gives 1plus integer terms, hence the first factor is A+Bi+Cj+Dkfor someA, B ,C ,D Z.The second factor is its conjugate, thus

p= A2 +B2 +C2 +D2 withA, B ,C ,D Z.

The following lemma is useful for showing that every odd prime is not a Hurwitzprime, which combined with Theorem 2.1, implies that every odd prime can beexpressed as the sum of four squares.

Lemma 2.2. If an odd prime p = 2n+ 1, then there are l, m Z such that pdivides1 +l2 +m2.

Proof. The squares x2, y2

of any two of the numbers 0,1,2,...,n are incongruentmod p because

x2 y 2 (mod p) x2 y2 0 (mod p)

(x y) (x+y) 0 (mod p)

x y or x +y 0 (modp)

x + y 0 (mod p) since 0


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6 JIA HONG RAY NG

By the four square identity (Theorem 1.3), every natural number is the sum of foursquares.

3. Discussion

The Four Square Theorem was first conjectured in Bachets 1621 edition of Dio-phantus and the first proof was given by Lagrange in 1770. Fermat claimed to havecome up with a proof, but did not publish it. In fact, there is a similar theoremcalled Fermats Two Square Theorem, that is a corollary of the Four Squares The-orem.

Theorem 3.1. Two Square Theorem: ifp = 4n + 1 is prime, thenp = a2 + b2 forsomea, b Z.

A proof exists that uses the unique prime factorization of Gaussian integersZ ={a+bi: a, b Z}and Lagranges lemma, a prime= 4n + 1 dividesm2 + 1for

somem Z.It has also been shown that only the numbers for which primes of the form 4k-1

do not appear to an odd power in their canonical decomposition can be representedas the sum of two squares. Thus, not all natural numbers can be expressed as thesum of two squares.

It is a similar situation for the sum of three squares, as it has been proved thatnumbers of the form 4n (8k+ 7) cannot be expressed as the sum of three squares.This proof is more involved than those for sums of two or four squares and is dueto the fact that a three square identity does not exist, i.e. the product of two sumsof three squares is not always a sum of three squares.

Waring generalized the question about the expression of numbers as the sum ofsquares to higher powers.

Theorem 3.2. Warings problem: There is a numberg (k) for every positive inte-gerk such that every integer can be written as at mostg (k) kth powers.

For example, g(2) is 4, g(3) is 9, g(4) is 19, g(5) is 37 and g(6) is 73. Euler

conjectured thatg (k) 2k+

32

k and Dickson, Pillai and Niven later conjectured

that the equality holds provided 2k

32

k+

32

k 2k, where {x} denotes the

fractional part of x. This equality has been proven to be correct for 6 n 471600000, but it is uncertain if it holds for all values ofn.

References

[1] J Stillwell. Elements of Number Theory. Springer-Verlag New York Inc. 2003.

[2] P Erdos, J Suranyi. Topics in the Theory of Numbers (2nd Ed). Springer-Verlag New YorkInc. 2003.

[3] I Niven, H S Zuckerman. Introduction to the Theory of Numbers (2nd Ed). John Wiley Sons,Inc. 1966.[4] Weisstein, Eric W. Warings Problem. From MathWorldA Wolfram Web Resource.

http://mathworld.wolfram.com/WaringsProblem.html

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